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The problem of fitting experimental data to a given model function $f(t; p_1,p_2,\dots,p_N)$ is conventionally solved numerically by methods such as that of Levenberg-Marquardt, which are based on approximating the Chi-squared measure of…

Least squares estimation, a regression technique based on minimisation of residuals, has been invaluable in bringing the best fit solutions to parameters in science and engineering. However, in dynamic environments such as in Geomatics…

The local least squares estimator for a regression curve cannot provide optimal results when non-Gaussian noise is present. Both theoretical and empirical evidence suggests that residuals often exhibit distributional properties different…

We address the numerical solution of minimal norm residuals of {\it nonlinear} equations in finite dimensions. We take inspiration from the problem of finding a sparse vector solution by using greedy algorithms based on iterative residual…

We propose and analyse a reduced-rank method for solving least-squares regression problems with infinite dimensional output. We derive learning bounds for our method, and study under which setting statistical performance is improved in…

Regression analysis is an important instrument to determine the effect of the explanatory variables on response variables. When outliers and bias errors are present, the standard weighted least squares estimator may perform poorly. For this…

We study the problem of choosing the best subset of p features in linear regression given n observations. This problem naturally contains two objective functions including minimizing the amount of bias and minimizing the number of…

We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates…

Partial least squares regression (PLSR) has been a popular technique to explore the linear relationship between two datasets. However, most of algorithm implementations of PLSR may only achieve a suboptimal solution through an optimization…

We study the following basic machine learning task: Given a fixed set of $d$-dimensional input points for a linear regression problem, we wish to predict a hidden response value for each of the points. We can only afford to attain the…

In the famous least sum of trimmed squares (LTS) of residuals estimator (Rousseeuw (1984)), residuals are first squared and then trimmed. In this article, we first trim residuals - using a depth trimming scheme - and then square the rest of…

The linear regression models are widely used statistical techniques in numerous practical applications. The standard regression model requires several assumptions about the regres- sors and the error term. The regression parameters are…

Additive regression models are actively researched in the statistical field because of their usefulness in the analysis of responses determined by non-linear relationships with multivariate predictors. In this kind of statistical models,…

The advantages and difficulties of application of Pad\'e approximants to two-dimensional regression analysis are discussed. New formulation of residuals is suggested in the method of least squares. It leads to a system of linear equations…

The least squares method provides the best-fit curve by minimizing the total squares error. In this work, we provide the modified least squares method based on the fractional orthogonal polynomials that belong to the space $M_{n}^{\lambda}…

We investigate the nonlinear regression problem under L2 loss (square loss) functions. Traditional nonlinear regression models often result in non-convex optimization problems with respect to the parameter set. We show that a convex…

Estimating linear, mean-square continuous functionals is a pivotal challenge in statistics. In high-dimensional contexts, this estimation is often performed under the assumption of exact model sparsity, meaning that only a small number of…

Shape-constrained convex regression problem deals with fitting a convex function to the observed data, where additional constraints are imposed, such as component-wise monotonicity and uniform Lipschitz continuity. This paper provides a…

We compute a \emph{sparse} solution to the classical least-squares problem $\min_x||A x -b||,$ where $A$ is an arbitrary matrix. We describe a novel algorithm for this sparse least-squares problem. The algorithm operates as follows: first,…

This book is meant to provide an introduction to linear models and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to ordinary least squares. In machine learning, the output is…